We extend the Rothschild and Stiglitz (1970) notion of increasing risk to families of random variables and in this way link their approach to the concept of stochastic processes which are increasing in the convex order. These processes have been introduced in seminal work by Strassen (1965), Doob (1968) and Kellerer (1972), who showed that such processes have the same marginals as a martingale. In fact, we demonstrate that their results include the results of Rothschild and Stiglitz as a special case. Further, we show that it makes sense to look at a larger class of processes, which we refer to as lyrebirds. We also show how these processes link up with the concept of second order stochastic dominance and are helpful in studying the dynamics of inequality and poverty measures. Further applications discussed include geometric and hyperbolic discounting, exotic derivatives and real options.